ELSEMER

ANNALS OF PURE AND APPLIED LOGIC

Annals of Pure and Applied Logic 79 (1996) 37-52

Systems of explicit mathematics with non-constructive p-operator. Part II

Solomon Feferman a1 *, Gerhard J2ger b* 1

’ Depnrfments of Mathematics and Philosophy, Stanford University, Stanford, CA 94305, USA

“lnstitut ,fir rnformatik and angewandte ~athematik~ Un~versit~t Bern, ~eub~ckstrasse IO, CH-3012 3ern,

Switzerland

Received 12 January 1995 Communicated by D. van Dalen

Abstract

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for ,u. The principal results then state: EET(p) plus set induction [type induction, formula induction) is proof-theoretically equivalent to Peano arithmetic PA (the second-order system (@,-CA) .+,, the second-order system (II:-CA) ,,,,,).

1. introduction

For the general background to this work, which continues the metamathematical study of systems of explicit mathematics introduced in [3], see the introduction of Part I [9], which treated applicative theories with (and without) the non-constructive (i.e. unbounded) minimum operator p. In this part we determine the effect of adding axioms for (variable) types (a.k.a. classes), together with several schemes for induction on the type N of natural numbers.

A base theory EET, called elementary explicit type theory, is introduced for this purpose. The schemes of induction on N considered are (S-I,) (set induction), (T-I,)

(type induction) and (F-I,) (formula induction). We determine the proof-theoretic strength of EET and EET(y) with each of these three schemes as follows:

IEEE + (S-I,) = PRA, EET(p) + (S-I,) E PA,

EET + (T-I,) = PA, EET(& + (T-I,) = (I-&CA),,,

EET + (F-I,) = (&CA), EET(p) -I- (F-I,) = (IIO,-CA)<,EO,

*Corresponding author. E-mail: sf~cs~i.stanford.edu. L E-mail: jaeger~iam.uni~.ch.

0168-~72/96/$15.00 0 1996 Elsevier Science BY. AI1 rights reserved SSDI 0168-0072(95)00028-3

38 S. Fcftirman, G. JtgerJAnnals of Pure und Applied Logic 79 (1996) 37-52

Thus the strongest of the systems considered is still weaker than full predicative analysis (@,-CA),..

The system EET provides a conceptual framework for the direct representation of notions from analysis. It was shown in [4] how extensive portions of (Bishop-style) constructive analysis can be formalized directly in a version EM,, 1 of EET + (T-I,). Correspondingly, it was indicated in [6] (see also [S]) how extensive portions of non-constructive classical and modern analysis can be formalized directly in a version W of EET(p) + (S-I,). Thus in both cases, systems of strength PA suffice for these portions of mathematical practice.

2. The syntax of applicative theories with types

The language L, of our applicative theories with types is a second-order extension of the first-order language L, of the basic theory of partial operations and numbers which is described at full length in [9]. We add to L, type variables and the binary relation symbol E for membership and a further binary relation symbol ‘8. The formula %(a,B) is used to express that the individual a represents the type B or is a name of B. Further new individual constants c, (e < o) for the uniform representa- tion of elementary types are available as well.

Variables. The language L, comprises individual variables a, b, c, x, y, z,f, g, h, . . .

and type variables A,B, C,X, Y,Z, . (both possibly with subscripts). Types are supposed to range over collections of individuals.

Constants. In addition there are the individual constants k,s (combinators), p,po, p1 (pairing and unpairing), 0 (zero), sN (numerical successor), pN (numerical predecessor), dN (definition by numerical cases), rN (recursion), p (unbounded min- imum operator) and c, (e < o), the meaning of which will be explained later.

The individual terms (r, s, t, ro, so, to, . . . ) of L, are generated as follows: (1) All individual variables and all individual constants are individual terms. (2) If s and t are individual terms, then so also is (s. t).

Thus the principal operation for the formation of individual terms is term application (s. t) which we often just write as (st) or st. In this simplified form we adopt the convention of association to the left so that s1 s2 . . . s, stands for ( . . . (sl . s2) . . . s,). In the following we write (tl, t2) for ptl t, and (tl, t2, . . . , t,) for (tl,(t2, . . . , t,)). Further we put t’ := sNt and 1 := 0’.

Relation symbols. The relation symbols of L,, are the unary 4 and N as well as the binary =, E and ‘R The relation symbols 1, N and = apply to individuals whereas E and ‘% represent relations between individuals and types.

The formulas (cp, x, $, cpO, x0, $o, . ) of L, are generated as follows: (1) tl, N(t), (s = t), (s E A), (A = B) and %(s,A) are (atomic) formulas. (2) If cp and $ are formulas, then so also are 1 q and (cp v $). (3) If 50 is a formula, then so also are (3x)q and (3X)(p.

S. Feferman. G. JCger,tAnnals of Pure and Applied Logic 79 (1996) 37-52 39

The remaining Iogical operations are defined as usual, An iL, formula q is called s~~u~~~e~ if the relation symbol % does not occur in cp; an U_, formula is called an elementary~ormula if it is stratified and does not contain bound type variables.

In the following we will make use of the logic of partial terms. Then tL is read “t is defined” or ‘“t has a value”. The partial equality relation E is introduced by

(s 2: t) := ((SJ, v tJ) + (s = t))

and (s # t) is written for (s.l A tl A 1 (s = t)). As additional abbreviations in connec- tion with the relation symbol N for the natural numbers we will use

t E N := N(t), AcN:=(vx)(xEA+xEN),

(3xEiV)4E):=(3~)(xENA~), (~xErV)y):=(~x)(xEM-,~).

Further, the fact that an individual term t plays the role of a total function from N” to N may be expressed as follows:

(1: N -+ N) := (Vx E N)(tx E N),

(r:N”+l -+ N) := (Vx E N)(tx: N” -+ N).

The logic of the applicative theories with types, which will be considered below, is the (classical) logic ofpartial terms due to Beeson [1], which is equivalent to the E+-logic with equality and strictness of Troelstra and van Dalen [l&J.

3. Elementary explicit type theory EET

Now we turn to the elementary explicit type theory EET. It is a second-order theory whose first-order part (axioms (l))(ll) below) corresponds to the theory BON of Feferman and Jager [9] and provides a series of axioms concerning the individuals and their applicative behaviour. We repeat them for completeness:

I. Partial combinatory algebra (1) kxY = X, (2) s.xpJ A sxyz CX XZ(YZ).

II. Pairing and projection

(3) (x9 y) L A Po(K Yf = x A Pi (X> Y) = Y, (4) (X,Y) f 0.

III. Natural numbers (5) 0 E A’ A (Vx E N)(x’ E N), (6) (vx E N)(x’ # OA p,(X’) = x),

(7) (‘dxE N)(x #O-+~,XENA(P~X)‘= x).

IV. De$nition by cases on N (8) a~N~b~Nr\a=b+dzvxyab=x, (9) aENAbENAa#b-+d,xyab=y.

40 S. Feferman. G. Jiigerl Annals of Pure and Applied Logic 79 (1996) 37-52

V. Primitive recursion on N

(10) (f:N+N)/\(g:N3+N)+(rNfg:N2+N),

(11) (f:N+N)A(g:N3-tN)~xENr\yENr\h=rhifg+hxO=fXAhX(y’)=

C?XY(hXY).

Hence the individuals form a partial combinatory algebra equipped with pairing

and unpairing, with natural numbers and their usual successor and predecessor

functions and with definition by numerical cases; rN acts as a recursion operator which

guarantees closure under primitive recursion.

Remark 1. It is mentioned in [9] that the possibility of lambda abstraction and the

recursion theorem follow from the axioms (1) and (2). Proofs of both results are given

in [3,1].

Now we turn to the second-order part of EET, i.e. we add axioms for types and

names. As mentioned above, !R(s, A) informally means that s represents the type A or

is a name of A. The internal representation of types by their names will be intensional

whereas types themselves should be considered extensionally.

VI. Extensionality

(Ed (Vx)(x E A ++x E B) -+ A = B.

In the following axioms about explicit representation and elementary comprehension

the relation symbol !R and the constants c, play a major role. The explicit representa-

tion axioms state that every type has a name. The constants c, serve to assign names to

all type terms in such a way that this assignment is uniform in the parameters which

occur in the type terms.

VII. Explicit representation

(E.1) (3x)Wx,A),

(W %(a, B) A %(a, C) -+ B = C.

It will be convenient for the naming process in connection with elementary compre-

hension, which is described below, to make use of the following conventions:

(1) We assume that there is some arbitrary but fixed standard assignment of Giidel

numbers to the formulas of [L,.

(2) We assume further that v0, v 1, . . and T/O, I/ 1, . . . are arbitrary but fixed

enumerations of the individual and type variables. If cp is an L, formula with no other

individual variables than 00, . . . , urn and no other type variables than T/O, . , Vn and if

<=x0,..., x, and 9 = Yo, . . . , Y,,, then we write cp[Z’,?] for the [I, formula which

results from cp by a simultaneous replacement of vi by xi and Vj by Yj (0 < i < m,

0 <j < n).

(3) Finally, if 2 = x0, . . . ,x, and 2 = X0, . . . ,X,, then %(Z,sr’) stands for

Al=0 s(Xi>xi).

S. Feferman. G. Jtger J Annals of Pure and Applied Logic 79 (I 996) 37-52 41

The following axioms depend on the GGdel numbering and the enumeration of the variables of il,. However, for obvious reasons this is not a serious restriction.

VIII. Elementary comprehension. Let cp[x,F,z] be an elementary IL, formula and e its Godel number; then we have

(ECA.l) (~x)(~x)(XEXH(p[X,~,i,Zj])

(ECA.2) !R(%,& @x)(x E A++cp[x,Z,i,]) -+ %(c,(Y$),A).

Hence the constants c, serve to provide names for the types which are generated by the formulas of the corresponding Godel numbers. This assignment of names to types is uniform in the individual and type variables which occur in the defining formula.

Remark 2. It is also possible to give a simple and very natural finite axiomatization of elementary comprehension. The basic idea is that one singles out a few operations, which are instances of (ECA.l) and (ECA.2), so that the full scheme can be obtained by combination of these operations. See the Appendix for more details.

The elementary explicit type theory EET is defined to be the [L, theory which consists of the individual axioms (l)-( 1 l), extensionality, the explicit representation axioms (E. 1) and (E.2) and the axioms about elementary comprehension (ECA. 1) and (ECA.2). An essentially equivalent formulation of EET was first introduced in [l 11;

a similar theory is presented in [S]. The first-order theory BON of [9] is the subtheory of EET which consists of the axioms (l)-(ll), restricted to the language L,.

4. Forms of induction and minimum operator

In this section we repeat the exact formulations of set and formula induction on the natural numbers presented in [9] and introduce the new form of induction - type induction - which is specific for second-order theories. In addition, the axioms for the non-constructive unbounded minimum operator p will be stated again.

First we recall from [9] that the subsets ofN, in contrast to the subtypes of N, are identified with the characteristic functions which are total on N. Accordingly P(N) is defined to be the type of all subsets of N, in the sense of characteristic functions, and we introduce the shorthand notation

u E P(N) := (Vx E N)(ax = 0 v ax = 1).

The three principles of complete induction on the natural numbers which will be considered later are the following.

Set induction on N (S-I,):

a~P(N)~u0=0~(Vx~N)(ux=0+ux’=0) -+ (Vx~N)(ux=0),

42 S. Feferman, G. JZger/ Annals qf Pure and Applied Logic 79 (1996) 37-52

Type induction on N (T-I,):

OEAA(V.XEN)(XEA-+X’EA) + (VXEN)(XEA)

Formula induction on N (F-I,):

~(0) A 0’~ E N)(cp(x) + cp(x’)) -+ (vx E NM4

for all formulas cp of I_,. Obviously (S-I,) can be regarded as a special case of (T-I,)

and (T-I,) as a special case of (F-I,). Adding these induction principles to the theory

EET yields the following new theories:

EET + (S-I,), EET + (T-I,), EET + (F-I,).

In the following we will show that the theory EET + (S-I,) is a conservative extension

of BON + (S-I,) and the theory EET + (T-I,) a conservative extension of BON +

(F-I,). In view of the results of Feferman and Jager [9] this means that EET + (S-I,)

is proof-theoretically equivalent to primitive recursive arithmetic PRA and

EET + (T-I,) to Peano arithmetic PA. We will also see that EET + (F-I,) is of the

same proof-theoretic strength as the theory (II:,-CA) of second-order arithmetic with

arithmetic comprehension.

Now we continue to follow the lines of [9] and add the non-constructive un-

bounded minimum operator /.L This functional assigns a natural number pLf to each

total function from N to N so thatf(pj) = 0 if there exists an x E N with the property

fx = 0. The exact axiomatization is as follows:

Axioms of the unbounded minimum operator:

(P.1) (f:N+N) + ,LL~EN,

(P.2) (f:N+N)~((3xcN)(fx=O) + f(pf)=O.

We shall write EET(,u) for EET + (~4.1, p.2). In this paper we provide a proof-theoretic

characterization of this system extended by (S-I,), (T-I,) and (F-I,).

Remark 3. In [12] a stronger form of the unbounded minimum operator p is

considered in which the axiom (cl. 1) is replaced by the equivalence

(f:N+N) c-f pfeN.

However, it turns out that this modification does not affect the proof-theoretic

strength of the theories considered in this article.

5. The proof-theoretic strength of EET and EET(p) with set and with type induction

Determination of the proof-theoretic strength of EET and EET(p) with set and type

induction on N is most simply established by a model-theoretic argument, showing

that these second order theories are conservative over suitable first-order extensions

S. Feferman, G. JCger J Annals of Pure and Applied Logic 79 (1996) 37-52 43

of BON. A careful analysis of the following arguments makes clear that they can be

formalized in suitable theories thus providing proof-theoretic reductions.

Lemma 4. Let A’ be a model of BON. Then there exists a model A* of EET which

has the following properties:

1. ,&‘* is an extension of A? in the sense that we have for all sentences cp of L,

J&*Fcp - JY+cp.

2. If .A! is a model of (F-I,) with respect to L,, then A?‘* is a model of (T-I,) with

respect to [L,.

Proof. Let &!’ be a model of BON and assume that M is the universe of ~2’. As a first

step we assign to each constant c, an element c, of M so that (i) there is no conflict with

the interpretation of the constants of L, in J&’ and (ii) c,v and i2,w are different in

./fi for all m # n and all v, w E M. Then, if T is a subset of the power set of M and

R a subset of M x T, we write (J&Z’, T, R) for the [L, structure in which the types range

over T, the relation symbol E is interpreted as the restriction of the usual element

relation to M x T and the symbols ‘$3 and ce are interpreted as R and i,, respectively.

The next step is to define, by induction on k < LO, subsets Rk of M together with

subsets ty(w) of M for all w E R,; furthermore we write Tk for {ty(w): w E Rk} : k = 0.

For every formula cp[x,y ] of L, with Godel number e and for all <E M we have

c,(;) E R, and set

ty( i?,(T)) := {m E M: A!’ k q[m,%]}.

k > 0: Rk contains Rk_ 1. In addition, for every elementary formula cp[x,y,z ]

of IL, with Godel number e and for all UE M and % E Rk_ 1 we have e,($G ) E Rk

and set

ty( i&( ;3,G)) := { meM: (~~,Tk-l,~)~=[m,~,ty(t;)l}

Here we use the shorthand notation that ty(G) stands for ty(w,), . . . , ty(w,) if 6J is

the sequence wl,...,w,.

Based on these definitions we now introduce collections T and R in order to

interpret the types and the representation relation:

T := u Tk and R := u {(w,ty(w)): w E Rk). k<w kio

It is then easily seen that .M * := (A, T, R) provides a model of EET which possesses

all the claimed properties. 0

Observe that the axioms of the unbounded minimum operator are first order.

Because of its first part, the previous lemma therefore also applies to BON(p) and

EET(,u). It immediately implies the following theorem which characterizes some

44 S. Feferman, G. JZgerl Annals of Pure and Applied Logic 79 (1996) 37-52

extensions of the second-order theories EET and EET( p) in terms of extensions of the

first-order theories BON and BON(p), respectively.

Theorem 5 (Conservative extensions). We have:

1. EET + (S-I,) is a conservative extension of BON + (S-IN);

2. EET + (T-I,) is a conservative extension of BON + (F-IN);

3. EET(,u) + (S-I,) is a conservative extension ofBON(p) + (S-IN);

4. EET(p) + (T-I,) is a conservative extension of BON(p) + (F-I,).

In view of the results of Feferman and Jager [9] we now obtain a proof-theoretic

characterization of some explicit type theories based on EET. In the following

corollary (IT:-CA) +, is the well-known system of second-order arithmetic for iter-

ated arithmetic comprehension through each ordinal less than s0 which is discussed in

detail for example in [2, lo].

Corollary 6. We have the following proof-theoretic equivalences:

EET + (S-I,) = PRA, EET(p) + (S-I,) = PA,

EET + (T-I,) = PA, EET@) + (T-I,) =(&CA)<,,.

Lemma 4 cannot be used, however, to analyse the theories EET + (F-I,) and

EET(p) + (F-I,). We cannot expect that the structures &‘* defined in Lemma 4

satisfy formula induction, even when ~fl does.

6. The proof-theoretic strength of EET and EET(p) with formula induction

The proof-theoretic strength of EET + (F-I,) has already been determined, among

other things, in [14] so that the corresponding result is stated here for completeness

only. The main focus in this section is on the proof-theoretic analysis of EET(p) +

(F-I,).

6.1. Lower bounds

The lower bounds of the theories EET + (F-IN) and EET( CL) + (F-IN) are easily

established by embedding suitable systems of second-order arithmetic in them. We do

not want to give a very detailed description of these systems now and confine

ourselves to mentioning the basic principles. All details are already presented in [9].

The language 3z of second-order arithmetic contains number variables (x, y, z, . . . ),

set variables (X, Y,Z, . ..). th e constant 0 and symbols for all primitive recursive

functions and relations. An 58; formula is called arithmetic if it does not contain

bound set variables, whereas free set variables are permitted.

The first such theory, which will be important for us, is the system (II:-CA) which

contains the usual machinery of primitive recursive functions and relations, the

S. Ft$&man. G. Jriger/Annals of Pure and Applied Logic 79 (1996) 37-52 4s

scheme of complete induction on the natural numbers for all LZz formulas plus the scheme of arithmetic comprehension for all arithmetic I”; formulas:

(~X)(~~}(X E xc-, q?(X)).

For defining the second theory, we refer to (initial segments of) the standard well- ordering < of order type r,, which is described, for example, in [15]. There one also finds the definition of the ordinal number E,~, which is crucial for us here. Finally, if IE is a natural number, then we write <, for the restriction of < to numbers the m < n, and ~~u~s~~~~e ~~~~~~~0~ Tl(a,cp) up to o! with respect to 40 is the formula

Wx + 4((VY i X)&Y) -+ cp(4)-+(‘dx < +P(x),

provided that the order type of the well-ordering <, is the ordinal a; for each CI < r, there is exactly one such n.

(II:-CA) _ is the extension of (II:-CA) obtained by adding axioms which state that the arithmetic comprehension axioms can be iterated through all ordinals less than a,,, plus transfinite induction Tl(x, q) for all CI < q. and all ?Zz formulas cp. More details on such second-order theories can be found in [2,9, lo].

Now we consider two forms of interpreting the language Zz of second-order arithmetic into IL,: In both cases the number variables of xl, are interpreted as ranging over N and the primitive recursive functions and relations are represented by indi- vidual terms of I,, which is possible by making use of the recursion operator rN. However, in the first case the set variables of LZ1 are interpreted as ranging over subtypes of N, whereas in the second case the set variables of _4u2 are interpreted as ranging over P(N).

Hence every _5?.. formula cp is translated according to the first interpretation into an [L, formula 40 ’ so that

((!lx)c;n(x)).‘.:= (3x E N)q’ (x),

This is in contrast to the second interpretation which translates the same yi; formula q into an II, formula (even an L, formula) qN which deals with the number and set variables according to

((3x)qB(x))N:= (3x E N)tpN(x),

((3 YIP(Y))N := (jY)(Y E WV A cpN(.Y1).

Observe that these two interpretations do not differ for arithmetic formulas of LPz without free set variables.

It is now easy to see that the theory (II>-CA) is contained in EET + (F-I,) via the first of these interpretations. EIementary comprehension in EET takes care of the scheme of arithmetic comprehension in (II%-CA), and all other steps are straight- forward.

46 S. Feferman. G. Jiigerl Annals of Pure and Applied Logic 79 (1996) 37-52

Proposition 7. We have for all LY2 sentences cp:

(&-CA) k cp =a EET + (F-I,) k- cp ’ .

Now we consider EET(,u) + (F-I,). It is a well-known result in proof theory that

(II:-CA) proves (‘VX)Tl(cc, X) for all CI < E,,,; cf. e.g. [ 151. Hence we may conclude

that EET + (F-I,) proves TZ(a, p) for all elementary I, formulas cp and all M < E,~.

Therefore the stage is set for embedding (I’I~-CA),,O into EET(p) + (F-I,). One

only has to follow the proof of Theorems 9 and 10 of [9] in order to make sure that

(the translation of) the arithmetic comprehension axioms can be iterated through all

ordinals less than eEo. The proof is exactly as in [9], with the only difference being that

in EET(p) + (F-I,) we have transfinite induction for elementary formulas for all

ordinals less than E,~, whereas in the theory BON(p) + (F-I,) of [9] it was available

only for all c( < Ed.

Proposition 8. We have for all -4”; sentences cp:

(&CA) <q,, t-- cp =, EET(d + (F-1,) I- ‘pN.

6.2. The theory &?

In this section we introduce the theory m of second-order arithmetic plus ordinals,

which will be used later to determine the upper bound for EET(,u) + (F-I,). The

theory I$& corresponds to the second-order version of the system PA,” of [9] with the

possibility to form subsets of the natural numbers by means of elementary compre-

hension. m and related systems are fully described and proof-theoretically analysed

in [13].

Let P be a new n-ary relation symbol, i.e. a relation symbol which does not belong

to the language _Yz. Then _Yz(P) is the extension of _Yz by P. An Yz(P) formula is

called P-positive if each occurrence of P in this formula is positive. We call P-positive

formulas without free or bound set variables, which contain at most 5? free, inductive operator forms, and let A(P,T ) range over such forms.

Y;, results from 6p2 by adding a new sort of ordinal variables (a, j, y, uO, DO, yO, . . . ), a new binary relation symbol < for the less relation on the ordinals’ and an (n + 1)-ary

relation symbol PA for each inductive operator form A(P,?) for which P is n-ary.

The number terms (s, t, sO, to, . . ) of -rP, are the number terms of _Yz, and theformulas (cp, $, x, qq,, I/I~, x0, . . . ) of _Yn are inductively generated as follows:

1. All atomic formulas of _Yz are atomic formulas of ZQ; further atomic formulas

of _Y* are (M < /3), (a = p) and P,(cr,Y)); we write PI(Y) for PA(a,T).

’ It will always be clear from the context whether < denotes the less relation on the nonnegative integers or on the ordinals.

S. Feferman, G. Jiger J Annals of Pure and Applied Logic 79 (1996) 37-52 47

2. If cp and $ are formulas of LG?~, then 1 cp and cp v $ are formulas of 9o.

3. If cp is a formula of L&, then (3x)(p, (Vx)cp, (3X)(p, (VX)cp, (3~)(p and (Vx)cp

are formulas of 9*.

If cp(P) is an _Y2(P) formula and $(X) an 6pn formula (where P is n-ary and2 is the

sequence x1, . . , x,), then q($) denotes the result of substituting $(7) for every

occurrence of P(Y) in q(P). For every L& formula cp we write (p3 to denote the

2& formula which is obtained by replacing all unbounded ordinal quantifiers (QP) in

cp by (Q/I < 2). Additional abbreviations are

P,‘“(Y) := (3fl< r)P:(T) and PA(Y) := (3~)Pi(y).

An _!& formula without free or bound set variables is called a At formula if all its

ordinal quantifiers are bounded; it is called a C* formula if all positive universal

ordinal quantifiers and all negative existential ordinal quantifiers are bounded;

correspondingly, it is called a II* formula if all negative universal ordinal quantifiers

and all positive existential ordinal quantifiers are bounded. The elementary -ri4, for-

mulas are the _Yfi formulas without bound set variables; free set variables, however, are

permitted in elementary formulas. See [13] for the precise definitions.

The theory m is formulated in the language -rZ; and contains the usual axioms of

predicate logic with equality plus the following non-logical axioms.

Number-theoretic axioms. These comprise the axioms of Peano arithmetic PA with

the exception of complete induction on the natural numbers.

Inductive operator axioms. For all inductive operator forms A(P,<):

P;(s)t,A(P;“,s).

IZ* rejection axioms. For every CR formula cp:

(CR-Ref) cp +(34(P”.

Linearity of the relation < on the ordinals

(LO) ~~ccA(~<pnB<~~r<y)A(Z</jVCI=pvP<~).

Elementary comprehension. For every elementary formula q(x) of _C$:

WA) (3X)Wly)(Y E X+-+&Y)).

Full induction on the natural numbers. For all SO formulas q(x):

W*-IN 1 440) A (~‘x)(cp(x) + CPW)) + W’M-4.

At induction on the ordinals. For all At formulas q(a):

(Ak) W’cc)((~‘p < 4cp(P) + V(U)) -+ W’cc)cp(4.

It follows from these axioms that m contains the theory PA”, of [9], However, the

following theorem, which is proved in [13], shows that m is significantly stronger

than PA:.

48 S. Fefetman. G. JiigerlAnnals of Pure and Applied Logic 79 (1996) 37-52

Theorem 9. m is proof--theoretically equivalent to @,-CA) +,,.

6.3. Upper bounds

In this section we concentrate on a proof of the fact that EET( p) + (F-IN) can be

embedded into m. As already mentioned, the theory EET + (F-I,) is studied in

[14], and the following result is proved there.

Proposition 10. EET + (F-I,) can be embedded into @IL-CA).

The main difference in the analysis of EET + (F-I,) and EET(p) + (F-I,) is the

way in which their applicative parts are modeled. The types and the type representa-

tion machinery are then treated in more or less the same manner. In case of the theory

EET + (F-I,) term application is interpreted in the sense of ordinary recursion

theory by translating the L, term (a. b) into {a}(b), where {e} for e = 0, 1,2, . . . is the

standard enumeration of the partial recursive functions.

In the presence of the unbounded minimum operator a more elaborate approach is

necessary, and the theories with ordinals turn out to be an adequate tool for handling

term application. In order to deal with the first-order part of EET( ,u) + (F-I,) we will

now follow Feferman and J;iger [9] and make use of the methods developed there.

First some standard notations of first- and second-order arithmetic: ( ... ) is

a standard primitive recursive function for forming n-tuples (to, . . , t,_ 1 ); Seq is the

primitive recursive set of sequence numbers; h(t) denotes the length of the sequence

number t; Seq, is the set of sequence numbers of length n; (t)i is the ith component of

the sequence number t, i.e. t = ((t)o, . . . , (t)lhcth_ 1 ); s E (X), stands for (s, t) E X.

Then we choose suitable numerals i, i, fi, &,, il, i&, cN, &, i, and i; as interpreta-

tions of the corresponding individual constants of L,. A further numeral i? is used to

interpret the individuals constants c, of [L, uniformly as (2, e). We also assume that

the individual variables and type variables of L, are properly mapped on (or simply

identified with) the number variables and set variables of &.

Since m contains PAZ, we know from [9] that there is a CR formula App(x, y, z) of

~5~ which is tailored for taking care of term application. Based on this formula, we

associate to each term t of [L, a Z* formula I/al,(x), which is inductively defined as

follows:

1. If t is an individual variable or an individual constant of L, and t^ the corres-

ponding term of TQ in the sense of the previous paragraph, then I/al,(x) is simply the

formula (t* = x).

2. If f is the individual term (TS), then Val,(x) is the formula

S. Fefennan. G. JCgerlAnnals of Pure and Applied Logic 79 (1996) 37-52 4’)

The translations ‘p* of elementary [L, formulas cp are then inductively defined as

follows:

1. For atomic formulas of fLP which do not contain type variables we put

(tl)* := (3x)Val,(x), (s = t)* := (3x)(Val,(x) A I/al,(x)),

Iv(t)* := (3x) I/al,(x), (t E x)* := (~x)(Val,(x) AX E X).

2. If cp is the formula 1 $, then (p* is 1 $*; if cp is the formula (Ic/ v x), then ‘p* is

(II/* v x*); if cp is the formula (3x)11/, then ‘p* is (3x)$*.

It follows immediately from the results of [9] that this translation of the first-order

part of [i, provides a sound interpretation of the first-order axioms of EET(p) into

%3.

Lemma 11. Zf cp is one of the axioms (l)-(11) of EET or one of the axioms of the unbounded minimum operator, then m proves ‘p*.

Now we turn to the second-order part of EET(p), which comprises types and the

explicit representation of types. The basic idea is to take the elementary definable

subsets of the natural numbers as types and to use the Godel numbers of their

definitions as representations. This has to be done carefully to ensure the strong

uniformity which is required in EET by elementary comprehension.

It follows from [9] that the formula App(x, y, z) can be chosen so that the individual

terms c,(Z) of [I,, which serve as explicit representation of types, are translated in m

as (t,e, (2)); i.e. we may assume that m proves

n-1

(s = c,(t(), . . . ) 4-l))* A vd(X)A /j v”&,(.Yi)+x = (ke,(Yo, . . ..Y.-i)). i=O

The *-translation of an elementary U, formula cp yields an elementary Zn formula ‘p*.

Hence standard arguments show that there is a truth definition in j% for the

elementary formulas of il,. However, this truth definition is not elementary.

Proposition 12. There exists an .LF* formula Tr(x, y, 2) so that m proves for every elementary II., formula cp [x0, . . , x,_ 1, Yo, . . . , Y, _ 1] and its Giidel number e:

Tr(e,x, Y)-cp*C(xL . . ..(?~).-~,(Y)~,...,(y)~-~l.

In the following we write Ele(e,m, n) for the primitive-recursive relation which

expresses that e is the Gijdel number of an elementary formula of [L, which has at

most m free individual variables and n free type variables. Following the pattern of the

proof of Lemma 4 we begin with introducing codes for the types. Obviously this can

be done in a primitive-recursive way.

50 S. Feferman, G. Jiiger 1 Annals of Pure and Applied Logic 79 (1996) 3 7-52

Proposition 13. There exists a primitive-recursive relation Code such that one can

prove in iI&:

Code(x) ++ Pe,s,m,n)We(e,m + l,n)~Seq,+,(s)

r\(Vi < n)Code((s),+i)~x = (e,e,s)].

Finally the Godel numbers of elementary formulas without type variables and at

most one free individual variable will determine our types. However, the comprehen-

sion principles of EET and the codes permit parameters so that an additional

consideration is necessary. The basic idea is to associate Godel numbers e which

satisfy Ele(e, 1, 0) to the above defined type codes.

Let f be an arbitrary unary primitive-recursive function. Then we introduce the

following abbreviation, which is convenient for formulating Proposition 14 below:

&r(e,s,X) := i

(!lm,n)[Ele(e,m + l,n)r\Seq,+,(s)~(Vi < n)Code((s),+i)

A(vi < n)Wx)(x E (Xh ~1 Tr(f((s),+i), <x>,O))l.

Iffis a function which assigns Godel numbers of formulas to codes, then dr(e, s, X)

means: (i) e is the Godel number of an elementary formula with m + 1 individual and

n type variables; (ii) s represents a sequence of m number parameters and n type codes;

(iii) the set X contains the information about the sets which are defined by the

formulas which correspond to these n type codes viaf:

By some straightforward but tedious manipulations with Giidel numbers and some

coding arguments it is possible to find primitive-recursive translations of type codes

into Gijdel numbers of appropriate formulas. More precisely:

Proposition 14. There exists a primitive-recursive function CO so that one can prove

in El%

1. Code(x) -+ Ele(@(x),l,O).

2. &(e,s,X) + CTr(~((ir,e,s)),(x),~) ++ Tr(e,(x)*s,X)l.

Now we are ready to tackle the second-order part of EET as well. First we

introduce two more definitions:

Rep(e,X) := Code(e)r\(~x)(x E X c) Tr(@(e), (x),0)),

Type(X) := (le)Rep(e,X).

The last step is now to extend the *-translation from the elementary formulas of U_, to

all formulas of [I,, which is inductively done as follows:

1. If cp is an elementary [L, formula, then ‘p* is already defined.

2. %(t,X)* := (3X)(Vd,(X)A k?p(X,X)).

3. If cp is a non-elementary formula of the form 1 II/, ($ v x) or (3x)$, then ‘p* is

i **, ($* v x*) or (3x)@*, respectively.

4. If cp is the formula (3X)$, then q* is the formula (3X)(Type(X) A cp*).

S. Feferman, G. Jci’ger/Annals of Pure and Applied Logic 79 (1996) 37-52 51

In view of Propositions 12-14 it follows that the translations of the explicit representation and elementary comprehension axioms are provable in m. Since full induction on the natural numbers is available in m, it is also clear that the translations of all instances of formula induction of EET do not create problems. Together with Lemma 11 we therefore obtain the following theorem.

Theorem 15. Let cp be a closed formula of [L,. Then we have

EET(p) + (F-I,) t- cp =s m t- ‘p*.

This was the last step which was missing for providing the proof-theoretic analysis of the theories EET + (F-IN) and EET(p) + (F-I,). The following result is an immediate consequence of the previous theorem and Propositions 7,8,10 and Theorem 9.

Corollary 16. We have the following proof-theoretic equivalences:

1. EET + (F-I,) = (&-CA). 2. EET(p) + (F-I,) = m = (IIO,-CA)<,r,,.

Appendix

Now we present a finite axiomatization EETf of EET. To this end we only have to show that the scheme (ECA), which consists of two parts (ECA.l) and (ECA.2), can be replaced by a finite number of its instances.

The language D_(EETf) of EETf is like the language of EET with the only difference that the infinitely many constants c, (e < w) are replaced by the following new individual constants: nat, id, co, int, inv and dom. - -

The axioms of EETf are the axioms of EET except the instances of (ECA) which are replaced by the universal closures of the following twelve axioms:

Natural numbers

6) PX)(vx)(x E xwN(x)), (ii) (Vx)(x E A c* N(x)) + !R(g, A),

Identity

(4 PX)(vx)(x E X-(~Y)((X = (Y,Y)))

(iv) WX)(XEA-(~Y)(X = (~,y)))-,Wid,A),

Complements

(4 (3X)(Vx)(x E X+--+x$% (vi) ~(b,B)r,(~x)(xEAHx~B)~~(COb,A),

Intersections (vii) (~X)(VX)(XEX++XEBAXEC),

(viii) %(b, B) A %(c, C) A (Vx)(x E A t-t x E B A x E C) + %(int (b, c), A), -

52 S. Feferman, G. Jtger JAnnais of Pure and Applied Logic 79 (1996) 37-52

Domains

(ix) PX)Wx)(x E X++(~Y)((X,Y) E B)),

(x) ~(b,B)A(~‘x)(xEXo(3y)((x,y)EB))~~(domb,A),

Inverse images (xi) (3X)(Vx)(x~XttaxEB),

(xii) %(b, B) A (Vx)(x E A -ax E B) -+ %(inv(a, b), A).

It is obvious that EETS can be embedded into EET. On the other hand it is also

possible to interpret EET into EETS. This is a consequence of the following proposi-

tion, whose proof is left to the reader as an exercise.

Proposition. Let cp[x,y,z] be an elementary formula of L(EET’) and e its Gtidel

number. Then there exists a term t of L(EET*), depending on e, so that we have:

1. (3X)(Vx)(x E X++cp[x,Z,Z]),

2. ~(~,~)*(~x)(XEAttcp[x,~,i,])~~(t(~,~),A).

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